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Dodecahedron in 3D

Twelve pentagons. Thirty edges. Twenty corners. Explore the regular solid, one face at a time.

On this page6 sections
01 / Explore the solid

A meeting of twelve pentagons

A regular dodecahedron with twelve flat pentagonal faces.12 faces · 30 edges · 20 vertices
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The regular dodecahedron. Every face is an equal, regular pentagon; all thirty edges have the same length.

One edge sets the size.

The view scales to fit. Units relabel values without converting them.

Volume

61.305units³

The space inside the solid
Surface area

82.583units²

The area of all twelve pentagons
See all measurements & formulas

What is a dodecahedron?

A dodecahedron is any three-dimensional solid with twelve faces. This is the regular dodecahedron: twelve identical regular pentagons, with three faces meeting at every vertex.

02 / The building block

Everything begins with a pentagon.

The geometry of a regular pentagon Five equal sides. Each interior angle is 108 degrees. A diagonal is the golden ratio times the side length. 108° diagonal = φa side a
φ = (1 + √5) / 2 ≈ 1.618, the golden ratio.

Five sides. Five equal angles.

Look straight at any face and you see the same regular pentagon. Each interior angle is 108°. Its diagonal is φ times the length of its side.

How many edges and vertices does a dodecahedron have?

03 / The unfolding

Dodecahedron net:
twelve pentagons.

Twelve pentagons stay joined along eleven folding edges. Follow the face numbers to see where each one belongs.

The dashed lines in this net are hinges. Fold them back together to close the dodecahedron. Although the net has more exposed boundary segments, the closed solid still has 30 edges: each edge joins two of the 12 × 5 pentagon sides.

Twelve numbered pentagons unfolded into a non-overlapping dodecahedron net. Dashed lines mark the eleven shared folding edges.
One possible net of a regular dodecahedron. No faces overlap when flat.
04 / Measurements

From one edge,
every measurement.

For a regular dodecahedron with
edge length a = 2 units.

Change the edge length ↑

Volume

Space enclosed by the solid

V = (15 + 7√5) / 4 × a³

(15 + 7√5) ÷ 4 × 2³

61.305units³

Surface area

Area of all twelve faces

S = 3√(25 + 10√5) × a²

3√(25 + 10√5) × 2²

82.583units²

One face

Area of a single pentagon

A = √(25 + 10√5) / 4 × a²

√(25 + 10√5) ÷ 4 × 2²

6.882units²

Inradius

Center to the middle of a face

r = √(250 + 110√5) / 20 × a

√(250 + 110√5) ÷ 20 × 2

2.227units

Circumradius

Center to a vertex

R = √3(1 + √5) / 4 × a

√3(1 + √5) ÷ 4 × 2

2.803units

Total edge length

Length of all thirty edges

E = 30a

30 × 2

60units

Results update with the size control and are rounded to three decimal places. These formulas apply to the regular dodecahedron.

The solid and its twelve faces.

A regular dodecahedron with numbered pentagonal faces and edge length a, alongside its twelve faces. Matching colors show the six pairs of opposite faces.
The same numbering and face colors are used in the model, the face index and the unfolding.
06 / Questions & facts

A closer look.

Fivefold symmetry
Viewed straight onto a face, the regular solid has rotational symmetry through 72°, one fifth of a full turn.
Angles between faces
The interior dihedral angle is approximately 116.565°. The 108° angle belongs to the flat pentagonal face.
Euler’s formula
Vertices − edges + faces = 20 − 30 + 12 = 2.
Are all dodecahedra made of regular pentagons?

No. The name describes a solid with twelve faces. The regular dodecahedron has twelve equal regular pentagons; other dodecahedra can have different face shapes.

How is it different from a dodecagon?

A dodecagon is a flat polygon with twelve sides. A dodecahedron is a three-dimensional solid with twelve faces.

What if I double the edge length?

Surface area multiplies by four and volume multiplies by eight. There are still twelve faces, thirty edges and twenty vertices.